#245217

order	512
nilpotency class	3
isoclinism	isoclinic class of #245204
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1190 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1190 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,8)(3,6,10,12)(4,7,11,14)(9,15,16,13)(17,18)(19,23)(20,30)(21,33)(22,26)(24,31)(25,36)(27,44)(28,38)(29,42)(32,43)(34,39)(35,41)(37,46)(40,47)(45,48);(6,12)(13,15)(17,19)(18,23)(20,27)(21,34)(22,28)(24,40)(25,42)(26,38)(29,36)(30,44)(31,47)(32,45)(33,39)(35,46)(37,41)(43,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,16)(14,15)(17,20,22,32)(18,24,26,41)(19,27,28,45)(21,35,36,31)(23,37,38,40)(25,43,33,30)(29,47,34,46)(39,48,42,44);(1,4,5,11)(2,7,8,14)(3,9,10,16)(6,13,12,15)(19,28)(23,38)(27,45)(29,34)(37,40)(39,42)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fi,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152413);